DAWSON COLLEGE – DEPARTMENT OF MATHEMATICSFINAL EXAM, FALL 2019

CALCULUS II (201-203-DW sections 01, 02, 03) DECEMBER 12, 2019 (9:30am-12:30pm) INSTRUCTORS: C. Farnesi, I. Rajput, A. Jimenez

INSTRUCTIONS:

The exam has 14 questions on 10 pages.

An information sheet is provided

Make sure “NOW” that all the pages are included with your exam. If thatis not the case, notify your teacher immediately

Do not detach any pages from this document. It should be returned intact.

NO MARKS are given for missing or improperly labeled answers

Only Sharp EL-531X – like calculators are allowed

1) Use the Riemann-sum approach to calculate the area under the graph of the function f (x)=2 x2

−x+2 over the closed interval [0,3 ] . [5 Marks]

2) Find y(x) given that dydx

=1+x+x2

√x and y(1) = 2 [5 Marks]

3) The number of subscribers S(t) to an online channel is increasing at the approximate rate of

S'(t)=5000−

1000(t+10)

2

Assuming this projected rate continues what will be the expected number of subscribers 10 months from now (t=10)? [5 Marks]

4) Find the average value of the following function over the [0, π] interval (use radians in the calculator) [6 Marks]

f (x)=sin (x)cos3(x )

5) Find the following indefinite integrals [24 Marks]

a) ∫x

(3−x )3dx (6 Marks)

b) ∫2x−1

(x+2)(x2+1)

dx (6 Marks)

c) ∫ x3 ln(x)dx (6 Marks)

d) ∫cos(ln x)

xdx (6 Marks)

6) Find the consumer's surplus if the demand function for the product isD(x)=84−5x2 , the supply function for the product is S(x)=x2

−12 and the current market price is set at the equilibrium price. [5 Marks]

7) Find the area of the region completely enclosed by the functionsf (x)=x2

−3x+7 and g (x)=4x−5 . [5 Marks]

8) Use the trapezoidal formula to calculate an approximate value of

∫1

22

√x+1dx with n=4 . Use 3 decimal places in your answer.

[5 marks]

9) Find the general solution y of the following separable differential equation.Do not isolate.

e2 y x3y '=x [5 Marks]

10) Calculate the following limit limx→0

3x2−3x+sin(3x)

3cos(2x)+3x−3ex [5 Marks]

11) Calculate the following improper integral [5 Marks]

∫1

∞ e−√ x

√xdx

12) Find the third Taylor polynomial of f (x)=√2x+4 at x=0 and use it to find an approximate value for f (0.5) . Use 5 decimal places in your answer. [5 Marks]

13) Does the series ∑n=0

∞ 5⋅3n+2−1

6n−1 converge or diverge? If it converges find its

value. [5 Marks]

14) Determine whether the following series converges or diverges. State each test used and explain each solution based on it [15 Marks]

a) ∑n=1

∞ n3−2n+1

5−n+4n3 (5 Marks)

b) ∑n=1

∞ n3√n5

−10(5 Marks)

c) ∑n=1

∞ lnn

n(5 Marks)